Electron beam dynamics in storage rings - Synchrotron · Lecture 10 - E. Wilson - 3/20/2008 - Slide...

Lecture 10 - E. Wilson - 3/20/2008 - Slide 1 1/29

Electron beam dynamics in storage ringsLecture 10 - MelbourneSynchrotron radiation

and its effect on electron dynamics

Lecture 10: Synchrotron radiation

Lecture 11: Radiation damping

Lecture 12: Radiation excitation


Summary of last lecture – Cavities II

• Conducting surfaces• Quality Factor, Filling Time and Shunt

Impedance• Corrugated structures• Dispersion in a waveguide• Iris loaded wavequide• Standing-wave and travelling wave structures.• Feeding, coupling and tuning structures.• Different modes


Contents

Introduction: synchrotron light sources

Lienard-Wiechert potentials

Angular distribution of power radiated by accelerated particlesnon-relativistic motion: Larmor’s formularelativistic motionvelocity || accelerationvelocity ⊥ acceleration: synchrotron radiation

Angular and frequency distribution of energy radiated: the radiation integralradiation integral for bending magnet radiationradiation integral for undulator and wiggler radiation


What is synchrotron radiationElectromagnetic radiation is emitted by charged particles when accelerated

The electromagnetic radiation emitted when the charged particles are accelerated radially (v ⊥ a) is called synchrotron radiation

It is produced in the synchrotron radiation sources using bending magnets undulators and wigglers


Why are synchrotron radiation sources important

Broad Spectrum which covers from microwaves to hard X-rays: the user can select the wavelength required for experiment

High Flux and High Brightness: highly collimated photon beam generated by a small divergence and small size source (partial coherence)

High Stability: submicron source stability

Polarisation: both linear and circular (with IDs)

Pulsed Time Structure: pulsed length down to tens of picoseconds allows the resolution of processes on the same time scale

Flux = Photons / ( s • BW)

Brightness = Photons / ( s • mm2 • mrad2 • BW )

synchrotron light


Brightness

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12

1.E+14

1.E+16

1.E+18

1.E+20

Brig

htne

ss (P

hoto

ns/s

ec/m

m2 /m

rad2 /0

.1%

)

Candle

X-ray tube

60W bulb

X-rays from Diamond will be 1012 times

brighter than from an X-ray tube,

105 times brighter than the SRS !

diamond


Applications

X-ray fluorescence imaging revealed the hidden text by revealing the iron contained in the ink used by a 10th century scribe. This x-ray image shows the lower left corner of the

page.

Biology

Medicine, Biology, Chemistry, Material Science, Environmental Science and more

A synchrotron X-ray beam at the SSRL facility illuminated an obscured work erased, written over

and even painted over of the ancient mathematical genius Archimedes, born 287 B.C.

in Sicily.

ArcheologyReconstruction of the 3D structure of a nucleosome

with a resolution of 0.2 nm

The collection of precise information on the molecular structure of chromosomes and their components can improve the knowledge of how the genetic code of

DNA is maintained and reproduced


Layout of a synchrotron radiation sourceElectrons are generated and accelerated in a linac, further

accelerated to the required energy in a booster and injected and stored in

the storage ring

The circulating electrons emit an intense beam of synchrotron radiation

which is sent down the beamline


Synchrotron Light Sources


Lienard-Wiechert Potentials (I)

The equations for vector potential and scalar potential

have as solution the Lienard-Wiechert potentials

with the current and charge densities of a single charged particle, i.e.

02

2

22

tc1

ερΦΦ −=

∂∂

−∇0

22

2

22 1

εcJ

tA

cA −=

∂∂

−∇

))((),( )3( trxetx −= δρ ))(()(),( )3( trxtvetxJ −= δ

ret0 R)n1(e

41)t,x( ⎥

⎦

⎤⎢⎣

⎡⋅−

=βπε

Φret0 R)n1(

ec4

1)t,x(A ⎥⎦

⎤⎢⎣

⎡⋅−

=ββ

πε

ctRtt )'('+=

[ ]ret means computed at time t’


The electric and magnetic fields generated by the moving charge are computed from the potentials

velocity field

Lineard-Wiechert Potentials (II)

retret Rnnn

ce

RnnetxE

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⋅−×−×

+⎥⎦

⎤⎢⎣

⎡⋅−

−= 3

0232

0 )1()(

4)1(4),(

βββ

πεβγβ

πε

&[ ]ritEn

c1)t,x(B ×=

and are called Lineard-Wiechert fieldstAVE∂∂

−−∇= AB ×∇=

acceleration field R1

∝ nBE ˆ⊥⊥r

Power radiated by a particle on a surface is the flux of the Poynting vector

BE1S0

×=μ ∫∫

ΣΣ Σ⋅=Φ dntxStS ),())((

Angular distribution of radiated power

22

)1)(( RnnSd

Pd β⋅−⋅=Ω

radiation emitted by the particle


Angular distribution of radiated power: non relativistic motion

Integrating over the angles gives the total radiated power

Assuming and substituting the acceleration field0≈β

( )2

02

22

0

2

)(4

1 βεπμ

&××==Ω

nnc

eERcd

Pdacc

2

0

2

c6eP βπε

&= Larmor’s formula

θβεπ

22

02

22

sin)4(

&c

ed

Pd=

Ω

ret

acc Rnn

cetxE

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ××=

)(4

),(0

βπε

&

θ is the angle between the acceleration and the observation direction


Angular distribution of radiated power: relativistic motion

Total radiated power: computed either by integration over the angles or by relativistic transformation of the 4-acceleration in Larmor’s formula

Substituting the acceleration field

Relativistic generalization of Larmor’s formula

( )[ ]

5

2

02

222

)1(

)(

41

β

ββ

επμ ⋅−

×−×==

Ω n

nn

ceER

cdPd

acco

& emission is peaked in the direction of the velocity

[ ]226

0

2

)()(6

βββγπε

&& ×−=c

eP

The pattern depends on the details of velocity and acceleration but it is dominated by the denominator


velocity || acceleration

Assuming ββ &|| and substituting the acceleration field

( )[ ]

( ) 5

2

02

22

5

2

02

2

)cos1(sin

c4

e

)n1(

)n(n

c4e

ddP

θβθ

επ

β

β

ββ

επΩ −=

⋅−

×−×=

&&

( )γ

ββ

θ211151

31arccos 2

max →⎥⎦

⎤⎢⎣

⎡−+=

62

0

2

c6eP γβπε

&=2

320

2

dtpd

cm6eP

πε=Total radiated power


velocity ⊥ acceleration: synchrotron radiationAssuming and substituting the acceleration fieldββ &⊥

( )[ ]

( ) ⎥⎦

⎤⎢⎣

⎡−

−−

=⋅−

×−×= 22

22

30

2

22

5

2

20

2

2

)cos1(cossin1

)cos1(1

c4

e

)n1(

)n(n

c4e

ddP

θβγφθ

θβεπ

β

β

ββ

επΩ

&&

When the electron velocity approaches the speed of light the emission pattern is sharply collimated forward

cone aperture

∼ 1/γ


Total radiated power via synchrotron radiation

2254

0

4

2

4

0

22

2

320

24

4

2

0

24

2

0

2

66666BE

cmece

dtpd

cme

EE

ce

ceP

o περγ

πεγ

πεβ

πεγβ

πε===== &&

Integrating over the whole solid angle we obtain the total instantaneous power radiated by one electron

• Strong dependence 1/m4 on the rest mass

• P(v ⊥ a) ≈ γ2 P(v || a)

• proportional to 1/ρ2 (ρ is the bending radius)

• proportional to B2 (B is the magnetic field of the bending dipole)

The radiation power emitted by an electron beam in a storage ring is very high.

The surface of the vacuum chamber hit by synchrotron radiation must be cooled.


Energy loss via synchrotron radiation emission in a storage ring

i.e. Energy Loss per turn (per electron) )()(46.88

3)(

4

0

42

0 mGeVEekeVUρρε

γ==

ργ

επρ 4

0

2

0 32 e

cPPTPdtU b ==== ∫

Power radiated by a beam of average current Ib: this power loss has to be compensated by the RF system

Power radiated by a beam of average current Ib in a dipole of length L(energy loss per second)

In the time Tb spent in the bendingsthe particle loses the energy U0

eTIN revb

tot⋅

=

)()()(46.88

3)(

4

0

4

mAIGeVEIekWP b ρρε

γ==

2

4

20

4

)()()()(08.14

6)(

mGeVEAImLLIekWP b ρρπε

γ==


The radiation integral (I)

Angular and frequency distribution of the energy received by an observer

[ ] 2

)/)((22

0

23

)1()(

44 ∫∞

∞−

⋅−

⋅−×−×

=Ω

dten

nnc

eddId ctrntiω

βββ

ππεω

&

Neglecting the velocity fields and assuming the observer in the far field: n constant, R constant

22

0

3

)(ˆ2 ωεω

EcRddId

=Ω

Radiation Integral

∫∫∞

∞−

∞

∞−

=Ω

=Ω

dttERcdtd

Pdd

Wd 20

22

|)(|ε

The energy received by an observer (per unit solid angle at the source) is

∫∞

=Ω 0

20

2

|)(|2 ωωε dERcd

Wd

Using the Fourier Transform we move to the frequency space


The radiation integral (II)

2

)/)((2

0

223

)(44 ∫

∞

∞−

⋅−××=Ω

dtennc

eddId ctrntiωβ

ππεω

ω

determine the particle motion

compute the cross products and the phase factor

integrate each component and take the vector square modulus

)();();( tttr ββ &

The radiation integral can be simplified to [see Jackson]

How to solve it?

Calculations are generally quite lengthy: even for simple cases as for the radiation emitted by an electron in a bending magnet they require Airy integrals or the modified Bessel functions (available in MATLAB)


Radiation integral for synchrotron radiationTrajectory of the arc of circumference

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛−=×× ⊥ θ

ρβε

ρβεββ sincossin)( ||

ctctnn

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛ ⋅− θ

ρβρωω cossin)( ct

ct

ctrnt

Substituting into the radiation integral and introducing ( ) 2/3223 1

c3θγ

γρωξ +=

In the limit of small angles we compute

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−= 0,sin,cos1)( tctctr

ρβ

ρβρ

( ) ⎥⎦

⎤⎢⎣

⎡+

++⎟⎟⎠

⎞⎜⎜⎝

⎛= )(K

1)(K1

c32

c16e

ddId 2

3/122

222

3/2222

2

20

3

23

ξθγ

θγξθγγωρ

επωΩ


Critical frequency and critical angle

Using the properties of the modified Bessel function we observe that the radiation intensity is negligible for ξ >> 1

( ) 11c3

2/3223 >>+= θγ

γωρξ

Critical frequency 3

23 γρ

ω cc =

Higher frequencies have smaller critical

angle

Critical angle3/11

⎟⎠⎞

⎜⎝⎛=ωω

γθ c

c

( ) ⎥⎦

⎤⎢⎣

⎡+

++⎟⎟⎠

⎞⎜⎜⎝

⎛= )(K

1)(K1

c32

c16e

ddId 2

3/122

222

3/2222

2

20

3

23

ξθγ

θγξθγγωρ

επωΩ

For frequencies much larger than the critical frequency and angles much larger than the critical angle the synchrotron radiation emission is negligible


Frequency distribution of radiated energyIntegrating on all angles we get the frequency distribution of the energy radiated

∫∫∫∞

=ΩΩ

=C

dxxKc

edddId

ddI

C ωωπ ωωγ

πεωω /3/5

0

2

4

3

)(4

3

ω << ωc

3/1

0

2

cc4e

ddI

⎟⎠⎞

⎜⎝⎛≈ωρ

πεω ω >> ωcc/

2/1

c0

2

ec4

e2

3ddI ωω

ωωγ

πεπ

ω−

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

often expressed in terms of the function S(ξ) with ξ = ω/ωc

∫∞

=ξ

ξπ

ξ dxxKS )(8

39)( 3/5 1)(0

=∫∞

ξξ dS

)(92)(

43

0

2

/3/5

0

2

ξεγ

ωω

πεγ

ω ωω

Sc

edxxKc

eddI

cc

== ∫∞


Frequency distribution of radiated energyIt is possible to verify that the integral over the frequencies agrees with the previous expression for the total power radiated [Hubner]

2

4

0

2

03/5

0

2

0

0

6)(

9211

ργ

εξξω

εγω

ω

ω

ξ

ω

ccedxxKd

ce

Td

ddI

TTUP c

bbb

==== ∫ ∫∫∞

The frequency integral extended up to the critical frequency contains half of the total energy radiated, the peak occurs approximately at 0.3ωc

50% 50%

3

23 γρ

ωε ccc

hh ==

For electrons, the critical energy in practical units reads

][][665.0][][218.2][ 2

3

TBGeVEm

GeVEkeVc ⋅⋅==ρ

ε

It is also convenient to define the critical photon energy as


Synchrotron radiation emission as a function of beam the energy

Dependence of the frequency distribution of the energy radiated via synchrotron emission on the electron beam energy

No dependence on the energy at longer

wavelengths


Polarisation of synchrotron radiation

In the orbit plane θ = 0, the polarisation is purely horizontal

Polarisation in the orbit plane

Polarisation orthogonal to the orbit plane

Integrating on all the angles we get a polarization on the plan of the orbit 7 times larger than on the plan perpendicular to the orbit

⎥⎦

⎤⎢⎣

⎡+

++

== ∫∞

22

22

2/5220

52

0

32

1751

)1(1

64e7d

ddId

dId

θγθγ

θγρπεγω

ΩωΩ

Integrating on all frequencies we get the angular distribution of the energy radiated

( ) ⎥⎦

⎤⎢⎣

⎡+

++⎟⎟⎠

⎞⎜⎜⎝

⎛= )(K

1)(K1

c32

c16e

ddId 2

3/122

222

3/2222

2

20

3

23

ξθγ

θγξθγγωρ

επωΩ


Synchrotron radiation from Undulators and wigglersPeriodic array of magnetic poles providing a sinusoidal magnetic field on axis:

),0),sin(,0( 0 zkBB u=

ztKctxtKtr uu

zuu ˆ)2cos(

16ˆsin

2)( 2

2

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛++⋅−= ω

πγλβω

πγλ mc

eBK u

πλ

20= Undulator

parameter

Constructive interference of radiation emitted at different poles

Solution of equation of motions:

λθλβλ

nd uu =−= cos

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

21

211

2

2

Kz γ

β

⎟⎟⎠

⎞⎜⎜⎝

⎛++= 22

2

2 21

2θγ

γλλ K

nu

n


Synchrotron radiation from undulators and wigglers

t1t2

Bending Magnet — A “Sweeping Searchlight”

Wiggler — Incoherent Superposition

Dipoles

t3 t4

(10-100) γ –1

t5

γ –1

Continuous spectrum characterized by εc = critical energy

εc(keV) = 0.665 B(T)E2(GeV)

eg: for B = 1.4T E = 3GeV εc = 8.4 keV

(bending magnet fields are usually lower ~ 1 – 1.5T)

Quasi-monochromatic spectrum with peaks at lower energy than a wiggler

Undulator — Coherent Interference

(γ N)–1

undulator - coherent interference K < 1

wiggler - incoherent superposition K > 1

bending magnet - a “sweeping searchlight”

2

2 212 2

u un

Kn nλ λλγ γ

⎛ ⎞= + ≈⎜ ⎟

⎝ ⎠2

2

[ ]( ) 9.496[ ] 1

2

n

u

nE GeVeVKm

ελ

=⎛ ⎞+⎜ ⎟

⎝ ⎠


Synchrotron radiation from Undulators and wigglers

For large K the wiggler spectrum becomes similar to the bending magnet spectrum, 2Nu times larger.

Fixed B0, to reach the bending magnet critical wavelength we need the harmonic number n:

Radiated intensity vs K: as K increases the higher harmonic becomes stronger

K 1 2 10 20

n 1 5 383 3015


Summary

Accelerated charged particles emit electromagnetic radiation

Radiation is stronger for circular orbit

Synchrotron radiation is stronger for light particles

Undulators and wigglers enhance the synchrotron radiation emission

Synchrotron radiation has unique characteristics


Bibliography

J. D. Jackson, Classical Electrodynamics, John Wiley & sons.E. Wilson, An Introduction to Particle Accelerators, OUP, (2001)M. Sands, SLAC-121, (1970)

R. P. Walker, CAS CERN 94-01 and CAS CERN 98-04K. Hubner, CAS CERN 90-03

J. Schwinger, Phys. Rev. 75, pg. 1912, (1949)B. M. Kincaid, Jour. Appl. Phys., 48, pp. 2684, (1977).

Electron beam dynamics in storage rings - Synchrotron · Lecture 10 - E. Wilson - 3/20/2008 - Slide...

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Transcript of Electron beam dynamics in storage rings - Synchrotron · Lecture 10 - E. Wilson - 3/20/2008 - Slide...